An approach for the measurement of the bulk temperature of single crystal diamond using an X-ray free electron laser

We present a method to determine the bulk temperature of a single crystal diamond sample at an X-Ray free electron laser using inelastic X-ray scattering. The experiment was performed at the high energy density instrument at the European XFEL GmbH, Germany. The technique, based on inelastic X-ray scattering and the principle of detailed balance, was demonstrated to give accurate temperature measurements, within \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$8\%$$\end{document}8% for both room temperature diamond and heated diamond to 500 K. Here, the temperature was increased in a controlled way using a resistive heater to test theoretical predictions of the scaling of the signal with temperature. The method was tested by validating the energy of the phonon modes with previous measurements made at room temperature using inelastic X-ray scattering and neutron scattering techniques. This technique could be used to determine the bulk temperature in transient systems with a temporal resolution of 50 fs and for which accurate measurements of thermodynamic properties are vital to build accurate equation of state and transport models.


Derivation of the fitting function
The detected scattered intensity, following the data processing described above, is expressed as the sum of the elastic coherent double differential cross section d 2 σ dΩdE el and the inelastic coherent one-phonon double differential cross section d 2 σ dΩdE inel 1 . These cross sections correspond to the probability of a photon being scattered elastically and inelastically by the sample in the solid angle Ω with energy between E and E + dE . One can write the instantaneous position, R s (t), of atom s in terms of the normal modes, or phonons, R s (t) = R eq s + u s (t). (1) R eq s represents the equilibrium position of atom s, and u s (t) is a harmonic displacement vector. Using the second quantisation of the vibrational modes in a solid in terms of the phonon creation and annihilation operators 1, 2 and assuming thermal equilibrium, u s (t) may be written as, In equation (2), M s is the mass of atom s, and N is the number of nuclei in the crystal. The summation over λ runs over all the phonon branches defined for the crystal (3 acoustic modes and 3(N b − 1) optical modes, with N b the number of atoms in the basis defining the crystal) and all phonon wavevectors, q, defined within the first Brillouin zone. ω λ ( q) is the frequency of a phonon characterised by branch λ with a wavevector q and a polarisation vector ε λ ,s ( q) in the first Brillouin zone. In the Heisenberg picture,â λ , q (t) =â λ , q exp{−iω λ ( q)t}.

Elastic coherent double differential cross section
The coherent cross section under a perturbation of the atomic position R s can be written Using equations (1), (2), and (3), the elastic coherent cross sections can be expressed as 1 where σ T is the Thomson coherent cross section for a single electron and Q the scattering vector defined as k out − k in with magnitude Q. V 0 the volume of the unit cell, H any reciprocal lattice vector andhω the energy exchanged during the scattering event. f s (Q) is the X-ray atomic form factor for atom s at Q, and 2W s (Q) is the Debye Waller factor associated with the element s. For a cubic crystal, 2W s (Q) is given by ω m,s is the maximum phonon frequency for element type s, Z s the normalised phonon density of states for element s. T the temperature of the lattice and k B the Boltzmann constant. Given the experimental configuration, equation (4) can be further simplified. First, as the measurement was made on diamond, there is only one element and so s takes only one value. Second, as the measurement was performed on a single crystal and Q was within the first Brillouin zone, H = 0. With these considerations, equation (4) becomes 2/6 and the prefactor I 0 is given by 1.2 Inelastic coherent double differential cross section Using equations (1), (2), and (3) the inelastic coherent cross sections can be expressed as.
Again, the summation over s runs through all the elements in the unit cell. n (ω λ ( q)) is the occupation number of phonons with frequency ω λ ( q). It is defined as n (ω)) = e¯h ωβ − 1 −1 with ω being a real positive number and β = 1 /k B T .
Given the experimental configuration, equation (8) can be simplified to Note that in equations (8) and (9), the energy differencehω is positive (negative) when the photon loses (gains) energy. However, for the rest of the discussion, we considerhω to be positive when the photon gains energy. As the probability of annihilating a phonon of frequency ω λ and hence transferring energy and momentum to the photon is given by n(ω λ ) , in our convention we should rewrite equation (9) as where the prefactors have been collected into a single term I λ given by In equation (10), the factors n (ω λ ( q)) + 1 and n (ω λ ( q)) are defined for positive frequency ω λ ( q). The equation can be re-written with the function n (ω) defined for positive and negative frequencies. To do this, we first utilise the definition of n (ω) Using this we can express equation (10) as Next, we use the sampling property of the Dirac delta function to write n(ω λ ( q)) δ (hω −hω λ ( q)) = n (ω) δ (hω −hω λ ( q)), (14a) n(−ω λ ( q)) δ (hω +hω λ ( q)) = n (ω) δ (hω +hω λ ( q)). (14b) These substitutions replace the values n(ω λ ( q)) with the function n(ω) which is independent of λ and hence can be taken out of the sum.
The previous derivation assumed that the displacement vector u s (t) is caused by harmonic forces. In reality, non-harmonic effects such as phonon-phonon interactions that may arise from terms beyond the second order in the Taylor expansion of the interatomic potential could play a role in the measured intensity. To account for such effects, the δ -Dirac function are replaced by a Lorentzian lineshape describing the damping of the phonon mode through non harmonic effects 2 , as Finally equations (6) and (16) are used to define the fitting function used to analyse the experimental data R(hω, q) is the instrument function as determined from the spectra obtained from 500 µm of PMMA for room temperature, and 250 µm SiO 2 for the heated case. In equation 17, T, I 0 , I λ , Γ λ ( q), Γ 0 , ω λ ( q) are kept as free parameters.

4/6 2 Determination of the energy resolution of the instrument
The inelastic scattering spectra used to determine the instrument function for the room temperature and heated (500 K) diamond measurements are shown in figure 1. The top row corresponds to the energy resolution as measured from the quasi-elastic scattering of 500 µm-thick PMMA sample for the left (a), centre (b) and right analyser (c). The bottom row shows the measurement for the heated case using 250 µm-thick SiO 2 sample. The energy resolution is defined as the Full Width at Half Maximum of the fitted Voigt profile (solid black line) to the recorded intensity. The vertical dashed lines correspond to the edges of the analyser and are described in the main manuscript.

Experimental results
Inelastic X-ray scattering spectra recorded on the centre analyser for room temperature diamond. The experimental data and the best fit (solid black line) are identical to the ones presented in figure 3 (b) in the main manuscript. As discussed in the main manuscript, the quasi-elastic component is not visible since the measurement was performed on high purity, single crystal diamond, see I 0 × Γ 0 values reported in table 2. The dot-dashed lines correspond to the Stokes and anti-Stokes components extracted for the best fit parameters reported in table 2. The asymmetry of these features is due to the asymmetry of the Bose-Einstein distribution. Counts per shot T = 303 ± 20 K Q = 0.51 ± 0.07 Å −1 Figure 2. Inelastic X-ray scattering data collected at room temperature on the centre analyser. The experimental data (blue squares) are shown along with the best fit using equation 1 see main manuscript (solid black line). The dot-dashed lines correspond to the inelastic components of the spectrum. The quasi-elastic component is not visible since its contribution is negligible due the single crystal nature of the sample. The asymmetry of the inelastic peaks is caused by the Bose-Einstein distribution. It should be noted that the experimental data are fitted directly using equation 1 (main manuscript), solid black curve, and then the Stokes and anti-Stokes features are then individually extracted after the fitting procedure.